Murphy is thought to have coined the legal term "wilful misconstruction" whilst representing the Donaldson v. Becket appeal to the House of Lords in 1774 against the perpetual possession of copyright. He died at Knightsbridge, London, and was buried at Hammersmith, London.

A biography was written in 1811 by Dr. Jesse Foote. Nathaniel Dance painted his portrait which is thought to now be in the Irish National Portrait Collection.

His elder brother used his mother's surname and was known as James Murphy French (1725–59). He lived in London with his brother.

Arthur Murphy was devoted to Ann Elliot and he missed her after her early death. After Arthur Murphy's death this relationship was written about by Fanny Burney who found Murphy's devotion to Eliot fascinating.[1]

Title page and scene from 1823 printing of The Citizen, produced in 1761

An example of Murphy's theatrical writings is The Citizen, a farce, first produced at Drury Lane in 1761. The play included Ann Elliot as Maria. Ann was Murphy's protege and mistress.[2][3]

Philpot, a wealthy skinflint, has bargained with Sir Jasper Wilding, a fox hunter, for his son Young Philpot, a buck and wastrel, to marry Maria Wilding, and for his daughter Sally to marry Wilding's son, for settlements and twenty thousand pounds paid to Sir Jasper. Young Philpot has lost a fortune, but borrows money from his father and embarks on an insurance fraud involving shipwrecked goods. Maria plans to marry Beaufort, who loves her. As Young Philpot tries to propose, she convinces him she is half-witted, and he spurns her. In the second act, Philpot senior is visiting Corinna, a lady of loose virtue, but hides under the table when his son calls upon her. He overhears as Young Philpot tells her how he has cajoled the money out of his father. Maria's brother surprises them, and old Philpot is also discovered, to their mutual shame. In the final scene Sir Jasper with a lawyer obtains Philpot's signature to the agreements, but meanwhile Maria, an educated girl, shows her strong character to Young Philpot and he again refuses to propose. Having signed away his rights old Philpot offers to marry her, but the lawyer reveals himself as Beaufort, and explains that he has swapped the deeds, so that Philpot has unwittingly signed his agreement for Maria to marry Beaufort.